Introduction

As described in Chapter 1, fracture (or failure) phenomena are observed over a very broad range of size scales, from atomistic-scale to microscopic-scale to macroscopic-scale fractures. A shear failure (or rupture) of laboratory-scale encompassed by continuum mechanics would be roughly of the order of 10–3 to 1 m. In contrast, shear rupture phenomena occurring in the Earth's interior, including microearthquakes and huge earthquakes, encompass a much broader range of size scales from 10–1 to 106 m. Rupture phenomena over such a broad scale range covering both laboratory-scale and field-scale encompassed by continuum mechanics, are characterized by scale-dependent physical quantities inherent in the rupture.

In general, physical quantities inherent in rupture phenomena can be categorized into two groups: scale-dependent physical quantities, and scale-independent physical quantities. As shown in previous chapters, the scale-dependent quantities include the breakdown zone length Xc, the breakdown time Tc (or its reciprocal fsmax), the nucleation zone length Lc, and the fault-slip acceleration D̈. Thus, it is an unavoidable fact that rupture phenomena are scale-dependent. As noted earlier, therefore, it is essential that the constitutive law must be formulated in such a way that the scaling property inherent in the rupture breakdown is incorporated into the law; otherwise, scale-dependent physical quantities inherent in the rupture over a broad scale range cannot be treated consistently and quantitatively in a unified manner in terms of a single constitutive law.